It's a song that's of its time.
I was 10 years old when it was current and, to me, it was just another novelty/comedy song like the others I heard during Summer Holidays on the BBC Light Programme(mostly on 'Housewives Choice'). My favourites were 'Right Said Fred' and 'Gossip Calypso' by Bernard Cribbins, 'That Noise' by Anthony Newley, and 'Transistor Radio' by Benny Hill. There were a lot more, many of them masterminded by one George Martin in his doldrum days at Parlophone before the Beatles came along.
It was also during this period that I first heard Flanders and Swann, a life-changing revelation for me and, I know, so many others.
So to return to the point, 'Little Boxes' never struck me then as a song of protest.
It does now, of course, but only as a song of very mild protest..